Recoverable Creep Deformation and Transient Local Stress Concentration Due to Heterogeneous Grain-Boundary Diffusion and Sliding in Polycrystalline Solids

ARTICLE IN PRESS

Journal of the Mechanics and Physics of Solids 56 (2008) 1460–1483 www.elsevier.com/locate/jmps

Recoverable creep deformation and transient local stress concentration due to heterogeneous grain-boundary diffusion and sliding in polycrystalline solids

Yujie Wei, Allan F. BowerÃ, Huajian GaoÃÃ

Division of Engineering, Brown University, Box D, Providence, RI 02912, USA

Received 8 June 2007; received in revised form 14 August 2007; accepted 15 August 2007

Abstract

Numerical simulations are used to investigate the influence of heterogeneity in grain-boundary diffusivity and sliding resistance on the creep response of a polycrystal. We model a polycrystal as a two-dimensional assembly of elastic grains, separated by sharp grain boundaries. The crystal deforms plastically by stress driven mass transport along the grain boundaries, together with grain-boundary sliding. Heterogeneity is idealized by assigning each grain boundary one of two possible values of diffusivity and sliding viscosity. We compute steady state and transient creep rates as functions of the diffusivity mismatch and relative fractions of grain boundaries with fast and slow diffusion. In addition, our results show that under transient conditions, flux divergences develop at the intersection between grain boundaries with fast and slow diffusivity, which generate high local stress concentrations. The stress concentrations develop at a rate determined by the fast diffusion coefficient, and subsequently relax at a rate determined by the slow diffusion coefficient. The influence of the mismatch in diffusion coefficient, loading conditions, and material properties on the magnitude of this stress concentration is investigated in detail using a simple model problem with a planar grain boundary. The strain energy associated with these stress concentrations also makes a small fraction of the plastic strain due to diffusion and sliding recoverable on unloading. We discuss the implications of these results for conventional polycrystalline solids at high temperatures and for nanostructured materials where grain-boundary diffusion becomes one of the primary inelastic deformation mechanisms even at room temperature. r 2007 Published by Elsevier Ltd.

Keywords: Recoverable creep deformation; Grain-boundary diffusion; Grain-boundary sliding; Transient stress concentration

1. Introduction

Grain-boundary (GB) diffusion and sliding are the dominant mechanisms of plastic deformation in polycrystalline metals and ceramics at high homologous temperatures. They also contribute to room- temperature plastic flow in nanocrystalline materials, where the fine grain size tends to suppress plastic flow by

ÃCorresponding author. ÃÃAlso corresponding author. E-mail addresses: [email protected] (A.F. Bower), [email protected] (H. Gao).

0022-5096/$ - see front matter r 2007 Published by Elsevier Ltd. doi:10.1016/j.jmps.2007.08.007 ARTICLE IN PRESS Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483 1461 dislocation motion while the high density of GBs accelerates diffusional creep. In addition, GB diffusion plays an important role in stress generation and relaxation in polycrystalline thin ﬁlms. GB diffusional creep in macroscopic polycrystals has been extensively studied. Nabarro (1948) and Herring (1950) ﬁrst pointed out that self-diffusion can cause crystals to change shapes and induce macroscopic plastic deformation at elevated temperatures. Coble (1963) studied the dependence of macroscopic strain rates _ on grain size l and applied tensile stress s due to collective GB diffusion and showed that

dDO s _ ¼ a 3 , (1) kBT l where a is a geometrical constant, D is the GB diffusivity and d is the thickness of a layer in which interface diffusion is supposed to take place; kB, T and O are the Boltzman constant, the absolute temperature and the atomic volume, respectively. Ashby and Verrall (1973) considered a geometrical model of creep flow taking into account strains due to GB diffusion and sliding, to reflect the fact that both GB diffusion and sliding plays an increasingly important role in the inelastic deformation of polycrystalline solids as the temperature rises or the grain size decreases. More recent work has extended the original Coble model by accounting for phenomena such as interface reactions (Ashby, 1969, 1972; Arzt et al., 1983; Cocks, 1992). In general, these models predict a nonlinear relationship between stress and strain rate. Numerical methods have been developed to study GB diffusion and associated mechanical behavior in polycrystalline solids. Needleman and Rice (1980) have first applied the finite-element method in modeling GB diffusion and creep. Pan and Cocks (1993) developed finite-element formulations for modeling GB diffusion in arbitrary networks of grains with straight GBs. They applied the numerical techniques, combined with a time integration algorithm, to study microstructure evolution during superplastic deformation in polycrystalline materials. Bower and Wininger (2004) extended the work of Pan and Cocks (1993) by using an advancing front algorithm to generate a sequence of adaptive, evolving finite-element meshes to solve the evolution of two-dimensional geometries. Sethian and Wilkening (2003) and Wilkening et al. (2004a,b) have used techniques from semigroup theory to study mass transport in microelectronic circuits due to electromigration and GB diffusion. GB diffusion is also an important stress relaxation mechanism in thin films based on experimental observations (Thouless et al., 1996; Kobrinsky and Thompson, 1998) and theories (Gao et al., 1999; Weiss et al., 2001; Guduru et al., 2003). Meanwhile, it has been observed to act in combination with other deformation mechanisms in thin films. For example, recent studies on the mechanical behaviors of polycrystalline thin films on substrates (Gao et al., 1999) have indicated that constrained GB diffusion1 induces crack-like singular stress fields which leads to novel dislocation mechanisms that are driven by locally induced, rather than globally applied, stresses. In situ TEM experiments (Balk et al., 2003), atomistic simulations (Buehler et al., 2003) and discrete dislocation simulations (Hartmaier et al., 2005) have shown that such constrained GB diffusion and the associated dislocation mechanisms dominate plastic deformation mechanisms in unpassivated films thinner than a few hundred nanometers. These studies are calling for broader investigations on the general importance of constrained GB diffusion mechanisms in polycrystalline materials. Coble-type GB diffusional creep, together with GB sliding accommodated by GB diffusion, have been considered as major deformation mechanisms in nanocrystalline materials by many authors (e.g., Gleiter, 1989). Although diffusion-controlled processes are typically activated only at high homologous temperatures, Yamakov et al. (2002) have argued, based on their molecular dynamics simulations, that Coble-creep should be a dominating mechanism in nanocrystalline materials even at low homologous temperatures. An important feature of plastic deformation controlled by GB diffusion is that it gives rise to a strain-rate sensitivity on the order of 1. Although strain-rate sensitivity around 1 is rarely observed in regular tensile and compressive tests for nanocrystalline materials, significantly enhanced strain-rate sensitivity has been reported in both h.c.p. and f.c.c. nc materials. Nc Cu synthesized by Jiang et al. (2006) showed a rate sensitivity about m ¼ 0:104 at room

1In columnar grain structures in thin ﬁlms on substrates, diffusion tends to generate stress singularities at the ends of GBs that terminate perpendicular to the substrate, if diffusion between thin ﬁlms and substrates are constrained. These stress singularities may lead to dislocation emission and initiate delamination. GB diffusion in such a situation is termed as ‘constrained GB diffusion’. ARTICLE IN PRESS 1462 Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483

1000

4 × 10-1/s 800 4 × 10-2/s

4 × 10-3/s 600 4 × 10-4/s

4 × 10-5/s Stress (MPa) 400

4 × 10-6/s 200 2 × 10-6/s

10-6/s 0 0 2 4 6 Strain (%)

Fig. 1. Stress–strain curves of brush-plated nanocrystalline Cu at different strain rates. Data replotted from Jiang et al. (2006). temperature in the strain rate range of 106=sto4 101=s, Fig. 1. The authors have attributed the high strain-rate sensitivity to the dominant unstructured high-angle GBs in the material. With the current technological trends in continuing miniaturization of structures in electronic devices and materials, a thorough understanding of the inelastic deformation mechanisms by GB diffusion and GB sliding is becoming increasingly important. On the other hand, existing experiments and theories have not been able to fully capture the complex interplay between different deformation mechanisms in GBs. In particular, the following issues are of interest for the present study.

Most existing theories have assumed homogeneous diffusivity in GBs. In reality, most material properties are heterogeneous at the scale of individual grains; high angle, relatively unstructured GBs usually have higher diffusivities compared to low angle, structured GBs. GB diffusivity in a polycrystalline solid could differ by several orders of magnitude from one grain to another. Constrained GB diffusion due to heterogeneous diffusivities can affect not only deformation mechanisms in GBs, but also those in grain interiors. Diffusional deformation in GBs is a kinetic process but existing theories are usually developed for steady- state creep which neglects transient effects in GB diffusion. On the other hand, a number of important phenomena, such as stress concentration at GB junctions, could occur in a transient time period and disappear at steady state. In such cases, further progress will be required for more rigorous modeling of stresses induced by heterogeneous GB diffusion. GB sliding accommodated by diffusion is not considered in most theoretical models; GBs are either assumed to slide freely or not allowed to slide at all. These two extreme cases cannot represent real deformation behaviors in GBs. Constitutive laws for creep in polycrystalline solids have been based on simple boundary-value problems. In the presence of heterogeneous diffusive behaviors at the scale of individual grains, there is strong coupling of localized deformation mechanisms which cannot be predicted by a macroscopic analysis. An example is the constrained GB diffusion and the associated dislocation mechanisms in thin ﬁlms (Gao et al., 1999; Balk et al., 2003; Buehler et al., 2003; Hartmaier et al., 2005).

Motivated by these issues, the objective of the present study is to address the mechanical behavior associated with heterogeneous GB diffusion and diffusion accommodated GB sliding. The plan of the paper is as follows. The computational method of investigation will be discussed in Section 2 and used to investigate deformation ARTICLE IN PRESS Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483 1463 behaviors induced by heterogeneous GB diffusion in Section 3. A generalized Maxwell model is adopted to interpret the numerical results in Section 4. Section 5 closes the paper with some concluding remarks.

2. Problem formulation and numerical method of investigation

We model a polycrystal as a two-dimensional assembly of elastic grains, separated by sharp GBs. The polycrystal is subjected to plane strain deformation with inﬁnitesimal strains. Deformation in grains is modeled with anisotropic elasticity and dislocation mechanisms are suppressed. The crystal deforms plastically by stress driven mass transport along the GBs, together with GB sliding. Heterogeneity is idealized by assigning each GB one of two possible values of diffusivity and sliding viscosity. We compute steady state and transient creep rates as functions of the diffusivity mismatch and relative fractions of GBs with fast and slow diffusion. In the following, we brieﬂy discuss the numerical method to investigate constrained GB diffusion. The reader is referred to Bower and Wininger (2004) for a detailed description of a more general formulation for modeling migration, diffusion, sliding of GBs as well as plastic deformation in grain interiors. The discussions below will focus on GB diffusion and GB sliding.

2.1. GB diffusion

For the GB shown in Fig. 2, the chemical potential per atom can be expressed as a function of time t and position s along the GB as (Herring, 1950)

mðs; tÞ¼m0 snðs; tÞO, (2) where snðs; tÞ denotes the normal traction at s, and m0 is a reference potential which can be taken as zero for the present study. Contributions from strain energy terms to the chemical potential are ignored in the above equation because they are small. The Nernst–Einstein equation for atomic ﬂux jðs; tÞ at s is dD qmðs; tÞ dD qs ðs; tÞ jðs; tÞ¼ ¼ n (3) kBTO qs kBT qs and Ojðs; tÞ gives the volumetric rate of diffusion in the GB for jðs; tÞ denoting number of atoms crossing unit length of the GB. Mass conservation requires that the rate of material accumulation at s is related to the divergence of volumetric ﬂux as qu qOjðs; tÞ n ¼ , (4) qt qs

j grain+ u+ u− σ n τ s t grain− n

Fig. 2. A GB G with displacement u and uþ at two sides, atomic ﬂux is denoted by j . Here (m, n) deﬁnes a local coordinate with n ¼ðn1; n2Þ along the GB normal and m ¼ðm1; m2Þ along the tangential direction, and t denotes the traction exerted on the surface belonging to grain . It has a normal part sn and a shear component ts, i.e., t ¼ snn þ tsm. ARTICLE IN PRESS 1464 Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483

þ where un ¼ðu u Þn is the width of new material being inserted at point s due to diffusion. Combining with Eq. (3), the above equation could be rewritten as qu dDO q2s ðs; tÞ n ¼ n 2 . (5) qt kBT qs Using u~n ¼ un=l; s~ ¼ s=l, t~ ¼ t=t0; s~ ¼ s=E , Eq. (5) can be normalized as 2 ~ qu~n ~ q s~ nðs~; tÞ ~ D ¼D 2 with D , (6) qt~ qs~ D0 3 where l is the averaged grain size, and t0 ¼ kBTl =dD0OE is a characteristic time controlling diffusional 2 process in a GB with diffusivity of D0, and E ¼ E=ð1 n Þ is the plane strain modulus. In contrast to D, which represents the diffusivity of any GB in a heterogeneous system, D0 is usually taken as a characteristic GB diffusivity in the system. Real physical data for D0, E, and n used in this paper are listed in Table 1.

2.2. GB sliding

There are two types of GB sliding. One of them is a thermally activated, rate-dependent process; the other may be referred as athermal GB sliding, which is a temperature independent process involving relative sliding of two GBs when the resolved shear stress overcomes a threshold resistance. The athermal GB sliding differs from thermally activated sliding in that it is essentially rate-independent. In our analysis, only thermally assisted GB sliding is taken into consideration. A Newtonian viscous type of GB sliding is assumed with qu Z s ¼ t . (7) qt s The dimensionless expression of Eq. (7), with u~s ¼ us=l, t~ ¼ t=t0 and t~s ¼ t=E , is given as 2 qu~s kBTl ¼ Z~t~s with Z~ . (8) qt~ dD0OZ

Table 1 Material parameters used for grain-boundary diffusion, grain-boundary sliding and grain interior deformation

Parameters for GB diffusion in Eq. (5) Frost and Ashby (1982)

23 NA Avogadro constant 6:022 10 =mol 23 kB Boltzman constant 1:38 10 J=K T Absolute temperature 500 K O Atomic volume of Cu 1:18 1029 m3 d Effective GB thickness involving diffusion 109 m 6 2 Dc Pre-coefﬁcient of GB diffusivity 5:0 10 m =s 3 Qv Activation energy for GB diffusion 1:04 10 J=mol l Grain size 100 nm D0 GB diffusivity of Cu Qv Dc exp NAkBT Parameters for GB sliding in Eq. (7) Z Viscosity to GB sliding Elastic constants for Cu (isotropic) E Young’s modulus 135 GPa n Poisson’s ratio 0.30

Elastic–plastic constants for Cu (anisotropic) Simmons and Wang (1971) C11 Elastic constant 170 GPa C12 124 GPa C44 75 GPa ARTICLE IN PRESS Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483 1465

The dimensionless coefﬁcient Z~ corresponds a normalized inverse viscosity and will be referred to as the normalized GB ﬂuidity. For calculations in this paper, we will assign one or two possible values of GB diffusivity D and sliding viscosity Z to GBs to study mechanical behaviors induced by heterogeneous GB properties.

2.3. Diffusion in GB junctions

In a microstructure composed of multiple grains, a junction of three or more grains is usually a site of high stress concentration due to severely mismatched elastic properties of adjacent grains, in which cavitation or damage could be initiated. In the scope of this paper, no damage or GB decohesion will be considered. GB junctions serve as bridges for atoms to transfer between interconnected boundaries. Mass conservation in the junction requires that the sum of ﬂuxes from all GBs meeting at the junction be zero. A strong constraint is applied in our numerical implementation to satisfy mass conservation and chemical potential continuity in any junction.

2.4. Finite-element implementation

For a body V with internal boundaries G, displacement discontinuities crossing boundaries may exist, as seen in Fig. 2. Let t be the traction along a GB and (vþ v) be the relative velocity between two sides of the GB (Fig. 2), the variation of generalized internal virtual power dP for the body is given as 2 Z Z dP ¼ r: de_ dV þ t ðdvþ dvÞ ds, (9) V G where dv is a kinematically admissible virtual velocity ﬁeld, (dvþ dv) is the jump in virtual velocity across the GB, and de_ is a virtual strain rate

1 T de_ ¼ 2 ð½rdvþ½rdv Þ. (10) To enforce displacement jumps due to GB diffusion in Eq. (6) and GB sliding in Eq. (8), we augment the virtual internal power principle with Lagrange multipliers dsn and dts (Bower and Wininger, 2004), corresponding to variations in normal and shear stress along G. They must be continuous along GBs but may have discontinuities across junctions. The internal virtual power can be rewritten as Z Z dP ¼ r: de_ dV þ t ðdvþ dvÞ ds VZ G Z Du q2s Du þ n þ n þ s ð Þ D 2 dsn ds Zts dts ds, 11 G Dt qs G Dt where the D preﬁx of a variable indicates its change from one time step to the next. We separate the variation of virtual internal power in Eq. (11) into two parts

dP ¼ dPv þ dPgb (12) with Z

dPv ¼ r: de_ dV (13) V representing the contribution by continuum elements in grain interiors and Z Z Z Du q2s Du ¼ ð þ Þ þ n þ n þ s dPgb t dv dv ds D 2 dsn ds Zts dts ds (14) G G Dt qs G Dt

2Here equations are written in dimensionless format. The ‘’ used in Eqs. (6) and (8) indicating dimensionless parameters is removed for convenience. ARTICLE IN PRESS 1466 Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483 being the variation of virtual internal power contributed by boundaries. Integrating Eq. (14) by parts and using Eq. (3), we obtain Z Z Z Z þ Dun qsn qdsn Dus dPgb ¼ t ðdv dv Þ ds þ dsnds D ds þ Zts dts ds |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflG G Dt G qs qs G Dfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}t X by GB elements þ jadsn . ð15Þ |{z}a2qG by junction elements Hence the variation of virtual internal power can be further divided into three parts

dP ¼ dPv þ dPG þ dPJ , (16) where Z Z Z Z þ Dun qsn qdsn Dus dPG ¼ t ðdv dv Þ ds þ dsnds D ds þ Zts dts ds, (17) G G Dt G qs qs G Dt X X dPJ ¼ jadsn þ jadsn (18) a2qGI a2qGO correspond to the contribution from continuum elements in grain interiors, GB elements, and elements in boundary junctions, respectively. Here qG denotes all ends of GBs, which includes both junctions in the body (qGI) and those in a sample’s surfaces (qGO). Unless stated otherwise, a symmetry boundary condition will be applied for GB junctions of the latter case, i.e., ja ¼ 0inqGO. The normal stress sn must be equal at the ends of the three GBs that meet at a triple junction for chemical potential continuity. This condition is enforced using a constraint. The coupled grain interior and GB deformation problem satisﬁes the following variational principle:

dP ¼ 0 (19) to ﬁrst order in dv,{dsn; dts}. Using the Galerkin method, the variations of internal power components, dPG and dPJ , can give the stiffness and residual for GB elements and junction elements, respectively. We have implemented the above numerical procedure in the commercial ﬁnite-element computer program ABAQUS/ Standard (Abaqus Reference Manuals, 2005) via user element subroutines. In the next section, we will apply the constitutive models and numerical scheme introduced above to study the effects of heterogeneity in GB diffusivity and sliding resistance on the creep response of a polycrystal.

3. Heterogeneous GB diffusion and GB sliding

In a polycrystal with plastic deformation exclusively accommodated by GB diffusion and sliding, these two processes occur concurrently. The macroscopic mechanical behavior of the solid is governed by the cooperation and competition between diffusion and sliding in individual GBs. To deﬁne a local measure of relative importance of diffusion versus sliding in a GB, we consider the ratio R between normalized GB diffusivity (Eq. (6)) and GB ﬂuidity (Eq. (8)), i.e., D~ dDOZ R ¼ ¼ 2 . (20) Z~ kBTl In a GB with R ¼ 1, for example, the local deformation rates by diffusion and sliding are comparable; while if R ¼ 0:01, the local deformation rate by GB diffusion is about two orders of magnitude slower than that by GB sliding. Macroscopically, GB sliding is expected to be the rate controlling mechanism if Rb1 (corresponding to fast diffusion or high viscosity to GB sliding), and GB diffusion is the rate-controlling mechanism when R51. Note that the GB sliding velocity is directly proportional to the local shear traction ARTICLE IN PRESS Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483 1467

σ 11 MPa 1000

750

500

250

3 1000 Steady-state 2.5 GB diffusion 500 Transient region 2 0 1.5 1 GB sliding Stress (MPa) -500 0.5 Loading -1000 Mechanism strain (%) 0 0 1 2 3 4 5 0 1 2 3 4 5 Strain (%) Strain (%)

Fig. 3. Representative deformation behavior in polycrystalline solids with GB diffusion and sliding: (a) the microstructure; (b) stress contour (s11) at a total tensile strain of 1% showing stress concentration at triple junctions during the transient period; (c) the stress–strain gbd curve showing a transient response followed by steady-state creep; (d) the evolution of mechanism strains by GB diffusion (e11 ) and GB gbs sliding (e11 ) during a loading cycle. The square markers indicate points of zero macroscopic stress.

(Eq. (8)) while diffusional creep is controlled by the gradient in local normal stress (Eq. (6)), with different characteristic relaxation times for each process. In order to illustrate the essential features of the mechanical behavior of a polycrystal deforming plastically by GB diffusion and sliding, we present first a representative simulation for a model of polycrystalline Cu with homogeneous GBs. In these simulations, uniform GB diffusion is assumed, i.e., D=D0 ¼ 1, D0 being the GB diffusivity for polycrystalline Cu at T ¼ 500 K. The viscosity of GB sliding is chosen to satisfy R ¼ 1. As a specific example, we simulate GB diffusion and sliding controlled plastic deformation in polycrystalline Cu with an averaged grain size3 of l ¼ 100 nm and at temperature T ¼ 500 K. The parameters used are listed in Table 1. The microstructure used is shown in Fig. 3a. Nodes along the left boundary are fixed in the horizontal direction and those along the bottom are fixed in the vertical direction. A 5% tensile strain is applied to the sample in the horizontal direction at a strain rate of 104/s, followed 4 by an unloading and subsequent compression at a strain rate of 10 /s. Contours of the stress component s11 at a tensile strain of 1% during the transient stage is shown in Fig. 3b. Note the high stress concentrations near triple junctions. The macroscopic stress–strain curve (computed from the relative displacement of the boundaries of the specimen, and the resultant forces applied to the boundaries) is given in Fig. 3c, which shows a transient period followed by steady-state creep. Our computations also allow us to calculate the contribution to the total plastic strain arising from the various processes that contribute to the deformation (we shall subsequently refer to these as ‘mechanism strains’). The plastic strain rates due to GB sliding and

3 2 The size of the ith grain is estimated by equating the area of the grain to pli =4. The averaged grain size l of a sample is the numerical average of li over all grains. ARTICLE IN PRESS 1468 Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483 diffusion can be computed from Z Z þ t gbd 1 ðDu Du Þn gbd gbd e_ij ¼ ninj ds; eij ¼ e_ij dt, (21a) V G Dt 0 Z Z þ t gbs 1 ðDu Du Þm ðnimj þ mjniÞ gbs gbs e_ij ¼ ds; eij ¼ e_ij dt, (21b) V G Dt 2 0 where V is the total volume of the sample and G is the network consisting of all GBs. Under steady-state conditions, the total strain rate is equal to the sum of these two quantities. Under transient conditions, there is an additional contribution to the strain rate from the elastic deformation within the grains. Fig. 3d gives the gbd gbs evolution of the strain rates along the loading direction (e_11 and e_11 ) due to GB diffusion and sliding. The rectangular markers in Fig. 3d indicate a zero macroscopic stress at that point. In what follows, we will further investigate steady state and transient creep rates as functions of the diffusivity mismatch and relative fractions of GBs with fast and slow diffusion.

3.1. Coble creep

In this subsection, we investigate Coble creep in a homogeneous polycrystal, in which all GBs have the same diffusivity and sliding resistance. In particular, our goal is to study the role of the relative rates of sliding and diffusion on the macroscopic creep behavior of the solid. When GB diffusion is the rate controlling mechanism in the material, stress s as a function of strain e can be expressed as a dimensionless function sdDO dDOE 3 ¼ f 3 e; R , (22) kBTl e_ kBTl e_ where e_ is the applied strain rate. For a ﬁxed R, it is expected that f will be a constant f ¼ 1=a at steady-state creep as predicted in Eq. (1). In the case that GB sliding is the rate controlling mechanism, a similar dimensionless equation could be written in the form of s G ¼ g e; R , (23) Zle_ Zle_ where G is the shear modulus. The microstructure shown in Fig. 4a is adopted to study the strain-rate dependence of stress in polycrystals whereby plasticity is exclusively accommodated by GB diffusion and GB sliding. A uniform diffusivity of D ¼ D0 is used and R ¼ 1 is chosen. Fig. 4b gives the stress strain curves at

10 10-4/s

8 Σ 7 × 10-5/s

4 × 10-5/s 4 × -5

Normalized Stress 3 10 /s 2 × 10-5/s 2

10-5/s 0 0 10 20 30 40 50 Normalized Strain

Fig. 4. Coble creep: (a) the adopted microstructure with 236 grains; (b) normalized stress–strain curves at different strain rates, S ¼ 3 5 3 s=ðkBTl e_=dDOÞ with e_ ¼ 10 =s. Strains for each curve are normalized by their respective characteristic strain kBTl e_=dDOE. ARTICLE IN PRESS Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483 1469

Σ 4 10 102

102 100 / D e D 100 10-2

-2 10-4

Normalized Steady-State Stress 10 -5 0 5 10-5 100 105 10 10 10 R R

102

0.55 GB diffusion 100 / D

e 0.5 D -2 10 strain rates GB sliding 0.45 Normalized mechanism 10-4 -5 0 5 0.4 10 10 10 -4 -2 0 2 4 R 10 10 10 10 10 R

3 5 Fig. 5. Influence of the GB diffusivity–fluidity ratio R: (a) the steady-state stress S versus R, where S ¼ sss=ðkBTl e_=dDOÞ and e_ ¼ 10 =s; (b) the ‘‘effective diffusivity’’ De versus R; (c) the fitted curve for R and De using Eq. (26); (d) the mechanism strain rates at steady state versus R (mechanism strains are normalized by the applied strain rate e_ ¼ 105=s).

3 different strain rates. Strain for each curve is normalized by kBTl e_=dDOE, here e_ is the respective strain rate 3 5 for each curve. Stresses in all curves have been normalized by kBTl e_=dDO with e_ ¼ 10 =s. The results show a clear dependence of stresses on strain rates, as in Eq. (1). Interestingly, the parameter a we obtained in the calculations is about 1, which is two orders of magnitude smaller than that given by Coble (1963) (a ¼ 148). There are two reasons for the discrepancy: ﬁrst, Coble’s result is based on a three-dimensional analysis of spherical grains and second, the Coble result assumes free GB sliding. For a regular array of freely sliding two- dimensional hexagonal grains, a ¼ 36 (Spingarn and Nix, 1978). In simulations with vanishing GB viscosity 5 5 (R ¼ 10 ) we obtain a ¼ 34 (Fig. 5b, De=D 34 at R ¼ 10 ), which is close to the result for freely sliding grains.4 With R ¼ 1, the viscosity of the GBs substantially reduces the rate of creep. The dependence of steady-state creep rate e_ on grain size l is self-evident by taking the right side of Eq. (22) to be a constant.

4It can be shown that the creep rate calculated with an average grain size of l is overestimated in a polycrystal with nonuniform grain size. To illustrate this with a simple model, consider N parallel dashpot elements, with the kth element representing the response of a grain 3 with a given grain size lk. Each element, based on Eq. (1), has a viscosity of Zk ¼ kBTlk=adDO Pwhere a is taken as a constant for a material of uniform grain size. Taking the macroscopic stress s as an average over all elements (s ¼ sk=N), we have

dDOs dDOs ¼ P P e_ a N 3 oa N 3 kBTðð1=NÞ k¼1lkÞ kBTðð1=NÞ k¼1lkÞ which suggests that a polycrystal with a random distribution of grain sizes should creep more slowly than a perfect array of hexagonal grains. Here we have used the inequality (Hardy et al., 1934) ! ! XN 3 XN 1 1 3 lk o l . N N k k¼1 k¼1 ARTICLE IN PRESS 1470 Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483

The collective plastic deformation by GB diffusion and sliding at grain level results in a macroscopic creep rate which can be written in the form of

dDeO s _ ¼ 3 , (24) kBT l where De is a macroscopic ‘‘effective diffusivity’’ representing contributions from both diffusion and sliding at the grain level. Indeed, experimentally measured GB diffusivity is an ‘‘effective diffusivity’’ since diffusion and sliding may need to cooperate for compatible deformation, especially when dislocation activities are suppressed. We will justify this conclusion at the end of this subsection. We will find De in a polycrystal with specific GB properties through numerical experiments. Using a simple formula for elastic–viscoplastic solids that s_ ¼ Eðe_ _Þ, together with Eq. (24), we can derive the following stress–strain behavior for a polycrystal with GB diffusion and sliding being the exclusive plastic deformation mechanisms, s dDeOE ¼ 1 exp 3 e , (25) sss kBTl e_ where 3 sss ¼ kBTl e_=dDeO denotes the corresponding steady-state stress as t !1. Once a stress–strain curve in a polycrystal with specific material properties is obtained, we can extract sss and De by fitting the stress–strain curve to Eq. (25). Indeed, a very good match is found between stress–strain curves from our numerical simulations and that described by Eq. (25) with appropriate sss and De. In the preceding calculations, we have assumed the GB diffusivity–fluidity ratio R ¼ 1. Now we take R as a variable while fixing the GB diffusivity. We vary the viscosity associated with GB sliding to identify the rate controlling mechanism in the polycrystal at different R. With the stress–strain curve from our calculation for a given R, we can determine sss and De by curve fitting using Eq. (25). Fig. 5a plots the relationship between 3 5 steady-state stress (normalized by kBTl e_=dDO with e_ ¼ 10 =s) and R, while Fig. 5b plots that between De and R. Interestingly, Fig. 5b shows a negative slop of 1 indicating a linear relationship between De and 1=R for R4102, indicating that GB sliding remains to be the rate controlling mechanism even when GB fluidity is two orders of magnitude larger than GB diffusivity. In this region, the ‘‘effective diffusivity’’ could be related to the uniform GB diffusivity D by De ¼ D=R. Further reducing GB sliding viscosity will shift the rate- controlling mechanism from GB sliding to GB diffusion. Our analysis suggests that the rate-controlling mechanism is GB sliding for R4102 and GB diffusion for Ro102. Despite the difference in R for the above calculations, the normalized mechanism strain rates shown in Fig. 5d indicate that GB diffusion and sliding play almost equally important roles in accommodating deformation at steady state. A qualitative understanding of the roles of GB sliding and diffusion in controlling the macroscopic stress in the polycrystal can be obtained by representing the deformation mechanisms as two parallel spring–dashpot elements, to be discussed in more detail in Section 4. Under steady-state conditions the stress in the two elements is given by ( Adsed diffusion element; e_ ¼ Asses sliding element; where sed and ses denote stresses in the diffusion and sliding element, respectively, and Ad and As are two coefficients that can be determined from the limiting cases of sliding and diffusion controlled deformation. Then the macroscopic response is 2A A e_ ¼ d s s, (26) Ad þ As where s ¼ðsed þ ses)/2 is the average macroscopic stress. The numerical results in Fig. 5b with R in the range 4 4 of (10 ; 10 ) can be fit by Eq. (26) with Ad ¼ 10, and As ¼ 0:1=R, as shown in Fig. 5c. Eq. (26) is able to represent the trends in Fig. 5b reasonably well. ARTICLE IN PRESS Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483 1471

The investigations in this section indicate that creep in a polycrystal requires cooperative GB sliding and GB diffusion. Such a constraint may explain the limited ductility of polycrystalline materials where dislocation mechanisms are largely suppressed and plastic deformation is mainly accommodated by GB diffusion and sliding, e.g., ceramics and nanocrystalline materials.

3.2. Scaling behavior by heterogeneous GB diffusion and sliding

We next turn to investigate the influence of heterogeneous diffusivity on the macroscopic mechanical behavior. We will show that the saturation in stress–strain curves that commonly occurs in Coble creep changes character if there is no continuous fast diffusion path across the GB network. Heterogeneous GB diffusion in polycrystals is somewhat analogous to bond percolation. The flow of atoms in GBs can be compared to the flow of electricity through a network of resistors represented by GBs. Atoms (electricity) diffuse easily along paths of high diffusivity (low resistors) but may be blocked by low diffusivity GBs (high resistors). If there is no trans-system diffusive path in the sample, the Coble-type creep may not be achieved in the time scale of interest. We consider the microstructure shown in Fig. 6a. GBs with a square marker have a 10 slow diffusivity of 10 D0 and the rest have D0. There is no cross-sample high diffusive path in this microstructure. The corresponding stress–strain curves at different strain rates, as shown in Fig. 6b, do not exhibit a Coble-type steady-state behavior. Mechanical deformation in a polycrystal with heterogeneity in GB diffusion and sliding can be better understood by defining a dimensionless parametric function sdDf O dDf OE Df 3 ¼ f 3 e; R; ; f , (27) kBTl e_ kBTl e_ Ds where Df =Ds is the ratio of fast diffusivity over slow diffusivity and f is the fraction of fast diffusion GBs. If we adopt Eq. (25) to describe overall stress–strain curves in the polycrystal, the ‘‘effective diffusivity’’ De may be represented by

De ¼ gðDs; Df ; f; RÞ, (28) where g is an undetermined function. The relationship between s and fDs; Df ; f; Rg is condensed to its dependence on De. The microstructure in Fig. 4a is used to construct the dependence of De on these parameters in function g. The inﬂuence of heterogeneity in GB diffusion and sliding on deformation in the microstructure has been investigated at three different cases: (a) Each GB is randomly assigned either a fast diffusivity Df or a slow 3 diffusivity Ds, with Df =D0 ¼ 1andDs=D0 ¼ 10 . The viscosity of GB sliding is taken to make R ¼ 1 in each

6000

5000 3 × 10-4/s 4000 3 × 10-5/s

3000 3 × 10-7/s

Stress (MPa) 2000

1000

0 0 1 2 3 4 5 Strain (%)

Fig. 6. Analogy between GB diffusion and percolation: (a) the adopted microstructure, GBs with a square marker may have a slow diffusion than others; (b) stress–strain curves at different loading rates. ARTICLE IN PRESS 1472 Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483 boundary, in which case GB sliding is the rate-controlling mechanism. (b) The same GB diffusivity as in (a) except R ¼ 104 in each boundary. In this case, GB diffusion is the controlling mechanism. (c) The same arrangement with the fast diffusivity GBs as in (a) but the diffusion in slow diffusivity GBs is completely turned off. A loading rate of e_ ¼ 105=s is applied to all calculations. Fig. 7 shows the stress–strain curves for different f for case (a). The results show that higher stresses and larger strains are required to reach steady-state diffusion as f decreases. (Similar behaviors exist for cases (b) and (c).) We then extracted the steady-state stress sss and ‘‘effective diffusivity’’ De by ﬁtting the calculated stress–strain curves to Eq. (25). Fig. 8a shows the steady-state stress versus f for the three cases. When GB sliding is the rate controlling mechanism (case (a) with R ¼ 1), there is no abrupt transition for steady-state stress as fast diffusion GB fraction f increases. When GB diffusion is the rate-controlling mechanism, there is a region around 0:4ofo0:7 with a quick transition from deformation controlled by fast diffusion GBs to that by slow diffusion GBs as f increases, resembling a percolating system. In case (c), some GBs are completely constrained and the system shifts from GB accommodated creep to purely elastic deformation below a critical diffusive GB fraction. Fig. 8b plots the ‘‘effective diffusivity’’ De versus f for the three cases. The inﬂuence of Df =Ds in Eq. (27) is also studied numerically. Fig. 9a shows the trends of steady-state stress versus Df =Ds for several f. Based on Eq. (25) and the known steady-state stress for each f, Fig. 9b shows the relationship between ‘‘effective diffusivity’’ De and Df =Ds for several f.

3.3. Transient stress concentration by heterogeneous diffusivity and sliding

In this section, we focus on the influence of heterogeneous diffusivity on the microscopic stress field in the polycrystal. The main observation in this subsection is that, if the diffusivity changes abruptly at a point in the GB network, an applied load on the body induces transient crack-like stress concentrations at the time scale controlled by the fast diffusivity, which subsequently relax out of the system at a rate governed by the slow diffusivity. Similar behavior will be shown in the presence of heterogeneous GB sliding. We consider the simple idealized microstructure illustrated in Fig. 10: the polycrystal is approximated as an assembly of rectangular grains, with GBs oriented parallel and transverse to the loading direction. The GBs that are normal to the loading axis have a position-dependent diffusivity as indicated in the figure: the region between 1oxo0 has a fast diffusion coefficient Df , while the region between 0oxo1 has a slow diffusion coefficient Ds. R ¼ 1 is maintained in the GB at any position. The vertical GBs are not modeled explicitly: instead, we assume that they act as perfect sources of flux, by enforcing the condition that the normal stress on the horizontal GB at x ¼1 equals zero. The solid may be subjected to either stress or strain loading in the vertical direction. Our main objective is to compute the variation of normal stress that acts on the horizontal GB.

20 φ = 32% φ =50% 15 Σ

φ =60% 10

Normalized Stress φ 5 =70%

φ = 100% 0 0 0.5 1 1.5 2 2.5 3 Strain (%)

3 Fig. 7. Stress–strain curves for different fractions of fast diffusion GBs f. Stresses are normalized as S ¼ s=ðkBTl e_=dDf OÞ with e_ ¼ 105=s. As f decreases, higher stresses and larger strains are required to reach steady-state creep. ARTICLE IN PRESS Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483 1473

105 101 -4 -3 -3 R = 10 , Df = D0, Ds = 10 D0 R = 1, Df = D0, Ds = 10 D0 Σ 104 100

103 -1 R = 1, Df = D0, Ds = 0 10 f 2 -2 10 / D 10 e D

1 -3 10 10 -3 R = 1, Df = D0, Ds = 10 D0

0 -4

Normalized Steady-state Stress 10 -4 -3 10 R = 10 , Df = D0, Ds = 10 D0 R = 1, Df = D0, Ds = 0 10-1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Φ Φ

Fig. 8. GB diffusion in a bi-diffusivity polycrystal (error bars are based on three calculations at a given f): (a) normalized steady-state 3 5 stress (S ¼ sss=ðkBTl e_=dDf OÞ with e_ ¼ 10 =s) versus fast diffusivity GB fraction f; (b) ‘‘effective diffusivity’’ De versus f.

100 103 Σ φ=1 φ=1.00 φ=0.70 φ=0.50 -1 102 10 φ=0.7 φ

=0.32 f φ=0.00 / D e D 1 φ=0 10 10-2 φ=0.32

Normalized Steady-State Stress φ=0.5 0 10 0 1 2 3 10 10 10 10 10-3 100 101 102 103 Df / Ds Df / Ds

3 5 Fig. 9. Inﬂuence of Df =Ds in a bi-diffusivity polycrystal: (a) normalized steady-state stress (S ¼ sss=ðkBTl e_=dDf OÞ with e_ ¼ 10 =s) versus Df =Ds for several f; (b) ‘‘effective diffusivity’’ De versus Df =Ds at several f.

In particular, we shall show that the jump in diffusivity and viscosity at x ¼ 0 generates a severe transient stress concentration. The normal traction s in the GB can be expressed in terms of a dimensionless function as s dDf OE Df Zh ¼ f 3 t; ; , (29) s1 kBTl Ds Zl where s1 is the applied stress and t is time. To avoid introducing an extra time scale in the system using strain controlled loading, we use stress 4 controlled loading to study the dependence of s on time t in Eq. (29). We assign Df =Ds ¼ 10 and Df =D0 ¼ 20 in the GB. A constant tensile stress of s1 is applied to the sample at t ¼ 0. The GB traction proﬁles at different snapshots are plotted in Fig. 11a. The history of normal traction in the GB shows four stages: (a) at t ¼ 0 the stress is uniform; (b) a stress peak develops at very early times near x ¼1 and propagates towards x ¼ 0; (c) the stress drops to a very low value in the region 1oxo0, and simultaneously a stress concentration develops near x ¼ 0 in the region 0oxo1; (d) the stress concentration gradually attenuates as ARTICLE IN PRESS 1474 Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483

. σ∞ (e∞)

x = 1 D x = -1 f Ds σ n = 0 σ n = 0 x

η u0- = u0+ η l h j0- = j0+ μ μ 0- = 0+

Fig. 10. A bi-diffusivity GB between two rectangular grains subjected to a constant strain rate e_1 or stress s1 in the far field . Diffusion along vertical GBs is assumed to be so rapid that sn ¼ 0 along the two ends of the horizontal GB, which represent the limit of infinite vacancy sources/sinks. Df and Ds are fast and slow diffusivities while Zh and Zl denote high and low viscosities. Displacement, flux and chemical potential at the junction are continuous.

6 3 t/t = 0.02

) s ∞

5 ) σ ∞ 2.5 / Df / Ds = 50 σ σ t/ts = 5 / 4 t = 0 σ D / D = 10 t 2 f s t/t = 0.01 D / D = 3 3 f 1.5 f s t/tf = 0.05 Df / Ds = 1 2 t/tf = 0.2 1

1 0.5

Normalized Stress ( 4 Normalized Stress ( Df / Ds = 10 0 0 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Normalized Position Normalized Position

Fig. 11. Normal traction profiles in the GB at different times for the sample in Fig. 10 subjected to a constant tensile stress of s1 ¼ 500 MPa from t ¼ 0: (a) transient stress concentration. The stress evolution has four characteristic periods, as detained in the text, 3 3 tf ¼ kBTl =E dDf O and ts ¼ kBTl =E dDsO; (b) Steady-state GB traction for several Df and Ds. the stress approaches the steady-state distribution. The two transients evolve with different characteristic time 3 scales: the characteristic time for stress evolution in the region 1oxo0istf kBTl =E dDf O, while that in 3 0oxo1 for relaxation to steady state is ts kBTl =E dDsO. Long term diffusion over the time scale of ts is similar to that in classical steady-state creep models. The steady-state stress profile is well described by the analytical solution given in Eq. (31) in Appendix A. With s1 ¼ 500 MPa, Fig. 11b plots several steady-state normal traction profiles for different ratios of Df =Ds based on Eq. (31). ARTICLE IN PRESS Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483 1475

Similar phenomena could be seen in the presence of heterogeneous GB sliding. We apply a constant shear 2 stress t1 at t ¼ 0 to the top surface of the sample. In the low viscosity region, Zl ¼ kBTl =dD0O and in the 3 high viscosity region, Zh ¼ 10 Zl. GB diffusivity in both regions is Df ¼ Ds ¼ D0. Fig. 12 plots shear traction profiles in the GB at different times, which resembles the normal traction profiles in the case of heterogeneous GB diffusion. The stress evolution shows three characteristic stages: (I) at very early times, relative high shear stress emerges over the entire boundary 1oxo1; (II) the stress drops to a very low value in the low viscosity region 1oxo0, and simultaneously an increasing stress concentration develops near x ¼ 0 in the region 0oxo1; (III) this stress concentration gradually attenuates as the stress approaches the steady-state distribution. The two transients evolve with different characteristic time scales: the characteristic time for stress evolution in the region 1oxo0istl Zll=G, while that in 0oxo1 for relaxation to steady state is th Zhl=G. The influence of the diffusivity ratio Df =Ds on stress concentration is investigated by varying Ds while keeping Df =D0 fixed at unity. Since most experiments are conducted using strain controlled loading, we apply a constant strain rate of e_ ¼ 104=s to the sample shown in Fig. 10. Fig. 13a plots the normal traction along the GB in which Df =Ds ¼ 100. Severe stress concentration due to heterogeneous diffusion is observed in Fig. 4 13b when Df =Ds ¼ 10 . Deformation in the low diffusivity GB cannot be accommodated by diffusion during the process, consequently giving rise to high stresses. As the difference in diffusivity in the two regions

t/th = 0.01 5 ) ∞ τ / t/th = 0.1 τ 4

t/th = 0.5 3 t/t = 0.1 l t/th = 5 t = 0 2 Normalized Stress ( 1

0 -1 -0.5 0 0.5 1 Normalized Position

Fig. 12. Shear traction proﬁles in the GB at different times for the sample in Fig. 10 subjected to a constant shear of t1 ¼ 500 MPa from 3 t ¼ 0. The stress evolution has three characteristic periods, as detailed in the text. Here tlZll=G and th ¼ Zhl=G, hence th ¼ 10 tl in this calculation.

2 2 0.5 0.5% 1.5 1.5 0.4

0.3 0.3% 1 1 0.2 0.5% 0.5% 0.1% 0.5 0.5 Normal Traction (GPa) 0.1 0.3% 0.3% Normal Traction (GPa) Normal Traction (GPa) 0.1% 0.1% 0 0 0 -1 0 1 -1 0 1 -1 -0.5 0 0.5 1 Normalized Position Normalized Position Normalized Position

Fig. 13. Normal traction proﬁles at e ¼ 0:1%, 0.3% and 0.5%. The bi-diffusivity GB is loaded at a strain rate of e_ ¼ 104=s. Diffusivity 2 4 2 2 Df =D0 ¼ 1 in the left half of the GB is ﬁxed. (a) Df =Ds ¼ 10 ; (b) Df =Ds ¼ 10 ; (c) Df =Ds ¼ 10 but e_ ¼ 10 =s. ARTICLE IN PRESS 1476 Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483

2 2 ) ) ∞ ∞ σ / σ / 1.5 1.8 σ max σ 1 1.6

0.5 1.4

0 1.2 GB Normal Traction ( Maximum Traction ( t/tf = 0.05 t/tf = 1 t/tf = 10 t/tf = 50 -0.5 1 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 Normalized Position Normalized Time (t/tf)

Fig. 14. Stress-ﬂuctuation in a polycrystal with bi-diffusivity in GBs. (a) GB normal tractions (normalized by the applied stress s1 ¼ 500 MPa) along the trajectory shown in Fig. 6a at several snapshots. (b) The evolution of maximum GB normal traction in the polycrystal.

becomes larger, the high diffusivity GB essentially becomes a crack over the characteristic time associated with diffusion on this boundary, inducing high stress concentration in the low diffusivity GB. Larger Df =Ds and/or Zh=Zl gives rise to higher s. Also, higher strain rates induce higher stress concentrations under the present assumption of no plastic deformation in the grain interiors, as indicated by Figs. 13a and c. These two calculations are performed with the same GB properties but at different loading rates. We further simulated stress evolution in a polycrystal sample with heterogeneous GB diffusivity. The structure shown in Fig. 6a is adopted. GBs with square markers have diffusivity of Ds and the 4 rest have Df , with Df =Ds ¼ 10 and Df =D0 ¼ 20. The diffusivity–fluidity ratio R ¼ 1 is maintained for each GB. A constant tensile stress s1 is exerted on the right side. Normalized GB normal traction along black arrows in Fig. 6b at several snapshots are shown in Fig. 14a. Small stress peaks develop in some fast diffusion GBs at very early stage (t=tf ¼ 0:05). These peaks dissipate quickly as loads are shifted to low diffusivity GBs. At the end, most loads are carried by the slow diffusion GBs. Note the high stress fluctuation along GBs. Fig. 14b plots the evolution of the maximum GB normal traction in the polycrystal. A peak at the early stage is observed in slow diffusion GBs. For such a large system, we have not taken the mesh size and time step sufficiently small to fully capture the transient stress concentration. In this sense, we expect a higher stress concentration if the mesh size and time step are taken to be smaller. The saturated peak stress indicates that the system is approaching steady-state creep in slow diffusion GBs. The transient stress concentration shown in the polycrystal is qualitatively consistent with the analysis for a single GB shown in Fig. 10. The behaviors of transient stress concentration for deformation accommodated by heterogeneous GB diffusion and sliding provide an explanation why nanostructured materials tend to be more ductile at lower strain rates (Cheng et al., 2005; Wei and Anand, 2007). If GB diffusion and sliding are dominant plastic deformation mechanisms, materials deformed at lower strain rates have lower risks of crack initiation and growth in GBs. In contrast, high strain rates tend to induce severe stress concentrations at GB junctions and give rise to higher risks of GB fracture, rendering the materials more brittle.

3.4. Recoverable creep deformation by diffusion and sliding

In this section, we will show that inelastic deformation by heterogeneous GB diffusion and sliding in polycrytalline solids can be partially recoverable. As seen in Section 3.2, there may be severe stress concentration in a polycrystal with heterogeneous GB diffusion and sliding viscosity. The strain energy associated with these stress concentrations can make a fraction of the plastic strain due to diffusion and sliding ARTICLE IN PRESS Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483 1477

500 II

400

300 I

200

Stress (MPa) Stress (MPa)

100 500

0 t (s) 0 0 0.2 0.3 2 IV III

0 0.2 0.4 0.6 0.8 1 Strain (%)

σ σ yy yy (MPa) (MPa) GB a 30% σ σ 600 n = 0 n = 0 150

GB b 70% 50 400

σ n = 0 0 GB c D s Df 200 σ = 0 σ = 0 n n -50

0 -100

1.5 1.5

t = 0.2

1 t = 0.2 1

t = 0.1 t = 0.1

0.5 0.5 t = 0.3 t = 0.3 Normalized Traction Normalized Traction

0 0 t = 0.4 t = 0.4

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Normalized Position Normalized Position

Fig. 15. Recoverable deformation in an idealized two-dimensional structure with heterogeneous GB diffusivity: (a) stress–strain curve for the structure shown in (b). The sample is subjected to a load–time proﬁle shown in the inset; (b) a structure with three GBs, Df and Ds represents fast and slow diffusion regions. Normal tractions nearby the two free surfaces in GB ‘a’ and GB ‘c’ and that at the middle point in GB ‘b’ is set to zero. Deformation and stress contour syy: (c) at ‘II’ marked in (b); (d) that at ‘III’; (e) normal traction in GB a at different snapshot, the four time sequences correspond to ‘I’ to ‘IV’ marked in (b); (f) normal traction in GB ‘b’. ARTICLE IN PRESS 1478 Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483

1600 4000

1200 4 R = 100 Df / Ds = 10 3000

800 2000 Stress (MPa) Stress (MPa) R = 10 400 Df / Ds = 100 1000 D / D = 1 f s R = 1 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Strain (%) Strain (%)

1 0.8 GB diffusion 0.8 0.6 0.6 GB sliding 0.4 0.4 GB diffusion 0.2 0.2 Mechanism Strain (%) Mechanism Strain (%) GB sliding 0 0 0 1 2 3 0 1 2 3 Total Strain (%) Total Strain (%)

Fig. 16. Recoverable creep behavior due to heterogeneous GB diffusion and sliding using the microstructure in Fig. 6b: (a) stress–strain curves for several Df =Ds; (b) stress–strain curves for different R; (c) evolution of mechanism strains during loading and unloading for the 4 case of Df =Ds ¼ 10 ; (d) mechanism strain evolution for R ¼ 100.

recoverable on unloading. A simple grain structure is chosen to show mechanical behavior by heterogeneous GB diffusion in individual GBs (Fig. 15b). The structure has three GBs, Df and Ds represents fast and slow diffusion regions, respectively. Normal tractions nearby the two free surfaces in GB ‘a’ and GB ‘c’ and that at the middle point in GB ‘b’ are set to be zero. A stress boundary condition is applied to the top surface with a load–time profile shown in the inset in Fig. 15a. Corresponding stress–strain behavior is shown in Fig. 15a. Stress contours syy at ‘II’ and ‘III’ marked in Fig. 15a are shown in Figs. 15c and d, respectively. The traction evolution with time in GB ‘a’ and GB ‘b’ are shown in Figs. 15e and f. During unloading, compressive stress develops in high diffusivity regions; the gradient of chemical potential in these GBs drives matter to diffuse out and recover a portion of plastic strain induced by GB diffusion. Recovery of inelastic strain is further investigated using a more realistic microstructure shown in Fig. 6a. GBs marked with a square have a diffusivity of Ds and all others have a diffusivity of Df ¼ D0. The sample is loaded at a constant strain rate of 105=s to a nominal strain of 2:5% and then unloaded at 105=s. We take the diffusivity–fluidity ratio R ¼ 1 for each GB. Stress–strain curves at different ratios of Df =Ds are shown in Fig. 16a. Mechanism strains contributed by GB diffusion and GB sliding (defined in Eq. 4 (21)), in the case of Df =Ds ¼ 10 , are shown in Fig. 16c. A significant fraction of inelastic strains contributed by both GB diffusion and GB sliding is recovered. Constrained GB sliding also can induce recoverable deformation. Here we assign the same diffusivity D0 to all GBs and increase R by increasing GB viscosity Z. Fig. 16b shows several stress–strain curves at different values of R. Mechanism strains contributed by GB diffusion and GB sliding are shown in Fig. 16d, for the case of R ¼ 100. Inelastic strain by GB diffusion is found to be substantially recovered. Rajagopalan et al. (2007) have recently studied deformation in freestanding f.c.c. nanocrystalline thin films and reported that ‘‘plastic’’ deformation in these films is recoverable. The evidence that heterogeneous GB diffusion can induce recoverable ‘‘plastic’’ deformation may ARTICLE IN PRESS Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483 1479

100 6 Df / Ds = 10 80 4 Df / Ds = 10 60 D D f Ds f σ σ = 0 n = 0 n 40 D / D = 102 Stress (MPa) f s 25 % 50 %

20 Df / Ds = 10

0 0 5 10 15 20 25 Time (s)

Fig. 17. Stress relaxation in a bi-diffusivity GB to show that stress relaxation is controlled by low diffusivity GBs. (a) The GB structure, GB diffusivity in the blue region is slower than those in the red regions. (b) Stress versus time for different values of Df =Ds. Stress relaxation is controlled by the slow diffusion region.

suggest that diffusion has played an important role in the observed recovery of ‘‘plastic’’ deformation in their experiments.

3.5. Characteristic time for stress relaxation is controlled by low diffusivity GBs

The microstructure shown in Fig. 17a is also used to investigate the characteristic time for stress relaxation under heterogeneous GB diffusivity, with Df =D0 ¼ 10. We apply displacement controlled boundary condition. The sample is loaded at a strain rate of 7:5 104=s in the ﬁrst 12 s. The boundary is then ﬁxed. Four simulation cases are conducted, with diffusivity in the middle half of the GB taken to be 5 Ds ¼ D0; D0=10; D0=100; D0=10 . Fig. 17 shows that stress relaxation is much slower in the presence of 6 heterogeneous GB diffusivity. For the case when Df =Ds ¼ 10 , there is almost no stress relaxation in the time interval of 12 s. The low diffusivity parts of GBs are the primary load carrying regions and control the characteristic time scale for stress relaxation. Meanwhile, the deformation is mainly accommodated by the high diffusivity GBs.

4. A generalized Maxwell model for recoverable deformation

In order to develop simple intuitive understanding for recoverable inelastic deformation induced by heterogeneous GB diffusion and sliding, we consider a generalized Maxwell model shown in Fig. 18a, in which a high viscosity element Zh corresponds to a low diffusivity GB and a low viscosity element Zl corresponds to a high diffusivity GB. One could easily derive the stress–strain relationship for a given loading rates, either in strain or stress loading conditions. 7 We have taken the spring stiffness K ¼ 40 GPa and dashpot viscosity Zh ¼ Zl ¼ 10 Pa s. The viscosity of 107 Pa s is obtained by matching the dimensionless equation for GB diffusion in Eq. (6) for polycrystalline Cu with l ¼ 100 nm and at T ¼ 500 K. A typical stress–strain curve for the structure under a constant strain rate of 1:5 103=s is shown in Fig. 18b, which resembles Fig. 3a obtained from polycrystalline aggregates with uniform GB diffusivity. Fig. 18c shows the overall stress–strain behavior as well as that in each element in the 8 7 structure subjected to stress loading with Zh ¼ 5 10 Pa s and Zl ¼ 10 Pa s. The sample is loaded at a stress- rate of 5 MPa/s and unloaded at 50 MPa=s. In the presence of heterogeneous viscosity, we see that the deformation is partially recoverable upon unloading. When the applied stress is fully removed, the element with low diffusivity is still in tension but that with high diffusivity is in compression. These internal stresses drive the structure to deform further while part of the strain is recovered, resembling the behavior shown in Fig. 15. ARTICLE IN PRESS 1480 Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483

1500

1000

500 K K 0

Stress (MPa) -500

η η h l -1000

-1500 0 2 4 6 Strain (%)

500

400 Combination of two

300

200 Low diffusive element 100 Stress (MPa)

-100 High diffusive element -200 0 0.5 1 1.5 Strain (%)

Fig. 18. A generalized Maxwell model is used to demonstrate that heterogenous diffusion could give rise to recoverable creep deformation. (a) The model. (b) The stress–strain relation for Zh ¼ Zl. (c) The overall stress–strain relation and that in the two elements for 8 7 Zl ¼ 5 10 and Zl ¼ 10 Pa s. The high viscosity element has low diffusion.

5. Conclusion

We have carried out a preliminary investigation of diffusional creep in polycrystalline solids with heterogeneous GB diffusivities. The main observations from our analysis can be summarized as follows.

Heterogeneous GB diffusion leads to time-dependent creep deformation which is partially recoverable over a characteristic time scale associated with diffusion on fast diffusion GBs. This recovery is driven by internal stresses induced by mismatches in GB diffusivity. Heterogeneous GB diffusion leads to severe stress concentration over the characteristic time scale associated with diffusion on high diffusivity GBs. However, over the long term, such stress concentration tends to attenuate as steady-state creep is approached. ARTICLE IN PRESS Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483 1481

GB diffusion and sliding have to cooperate to allow compatible deformation ﬁeld in polycrystals. A macroscopic creep rate is controlled by the slower processing of these two mechanisms. At steady state, each mechanism contributes half of the macroscopic creep rate. In polycrystals with bi-diffusivity, the deformation resembles that in a percolating system when GB diffusion is the rate-controlling mechanism. The transition from fast diffusion-controlled deformation to that by slow diffusion occurs in the region 0:4ofo0:7asf decreases, here f is the fraction of fast diffusion GBs. When deformation in slow diffusion GBs is completely constrained, an abrupt transition from GB accommodated creep to elastic deformation is seen at f 0:6. Deformation is mainly accommodated by high diffusivity GBs while stress is primarily carried by low diffusivity GBs. On an intermediate time scale before steady-state creep is reached, high diffusivity GBs behave like cracks and result in severe stress concentration in low diffusivity GBs. The characteristic time scale controlling stress relaxation is determined by low diffusivity GBs.

Although GB diffusion is considered as the primary mechanism of mass transport in our analysis, dislocation emission–absorption in GBs can lead to additional mass transportation in conventional polycrystalline solids at high temperatures and for nanostructured materials even at room temperature. Such a scenario raises an interesting question of how GB diffusion is coupled to dislocation emission–absorption. GB migration can also inﬂuence mass transport and deformation in nanostructured materials, especially during thermal cycling. The coupling of various alternative and competing mechanisms including GB diffusion, GB migration, dislocation emission/absorption and GB sliding should play important roles in the deformation and reliability of small structures, e.g., metallic interconnects in electric circuits and nanocrystalline materials. Further studies of the coupling among different mechanisms are left to future work.

Acknowledgments

The research reported has been partially supported by the NSF MRSEC Program at Brown University under Award DMR-0520651. This support is gratefully acknowledged.

Appendix A

The analytical solutions for steady-state creep along a bi-diffusivity GB shown in Fig. 10 can be easily derived as follows. The diffusion equations in the two regions of the GB are 8 > 2 > q s < Df for xo0; qu qx2 ¼ e_ ¼ qt > q2s :> D for x40: sqx2 At steady state, the left side of the above equations is constant, and the general solutions can be written as ( 2 a1ðx þ 1Þ þ b1ðx þ 1Þþc1 for xo0; s ¼ 2 a2ðx 1Þ þ b2ðx 1Þþc2 for x40 with boundary conditions: s ¼ 0atx ¼1, (30a)

s ¼ 0atx ¼ 1, (30b)

j0þ ¼ j0 at x ¼ 0 flux continuity, (30c)

u0þ ¼ u0 at x ¼ 0 displacement continuity, (30d) ARTICLE IN PRESS 1482 Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483

m0þ ¼ m0 at x ¼ 0 chemical potential continuity, (30e)

2e_ ¼ jx¼1 jx¼1 mass conservation, (30f) Z 1 1 s1 ¼ s dx force balance. (30g) 2 1 The normal traction proﬁle along the GB can therefore be derived as 8 > e_ 2 e_ð3Df þ DsÞ <> ðx þ 1Þ þ ðx þ 1Þ for xo0; 2Df 2Df ðDf þ DsÞ s ¼ (31) > e_ 2 e_ðDf þ 3DsÞ :> ðx 1Þ ðx 1Þ for x40 2Ds 2DsðDf þ DsÞ and 2 2 Df þ 14Df Ds þ Ds s1 ¼ e_. (32) 24Df DsðDf þ DsÞ The maximum traction occurs at

Df Ds xm ¼ (33) 2ðDf þ DsÞ with value equal to 2 Df ðDf þ 3DsÞ smax ¼ 3s1 2 2 . (34) ðDf þ DsÞðDf þ 14Df Ds þ Ds Þ The steady-state solution for the bi-diffusivity GB can be reduced to that of a GB with uniform diffusivity5 if Df ¼ Ds ¼ D0. At steady state, the stress concentration factor induced by heterogeneous GB diffusion for the sample Fig. 10 is smax=s1 ¼ 3ifDf bDs.

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5 2 For the same structure in Fig. 10 with uniform GB diffusivity, steady-state normal traction within the GB is s ¼3s1ðx 1Þ=2 and e_ ¼ 3s1D0. ARTICLE IN PRESS Y. Wei et al. / J. Mech. Phys. Solids 56 (2008) 1460–1483 1483

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